3.464 \(\int x^2 (a+b \log (c (d+e x^{2/3})^n)) \, dx\)

Optimal. Leaf size=130 \[ \frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac{2 b d^2 n x^{5/3}}{15 e^2}-\frac{2 b d^4 n \sqrt [3]{x}}{3 e^4}+\frac{2 b d^3 n x}{9 e^3}+\frac{2 b d^{9/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{3 e^{9/2}}+\frac{2 b d n x^{7/3}}{21 e}-\frac{2}{27} b n x^3 \]

[Out]

(-2*b*d^4*n*x^(1/3))/(3*e^4) + (2*b*d^3*n*x)/(9*e^3) - (2*b*d^2*n*x^(5/3))/(15*e^2) + (2*b*d*n*x^(7/3))/(21*e)
- (2*b*n*x^3)/27 + (2*b*d^(9/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/(3*e^(9/2)) + (x^3*(a + b*Log[c*(d + e*x
^(2/3))^n]))/3

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Rubi [A]  time = 0.0780661, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2455, 341, 302, 205} \[ \frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac{2 b d^2 n x^{5/3}}{15 e^2}-\frac{2 b d^4 n \sqrt [3]{x}}{3 e^4}+\frac{2 b d^3 n x}{9 e^3}+\frac{2 b d^{9/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{3 e^{9/2}}+\frac{2 b d n x^{7/3}}{21 e}-\frac{2}{27} b n x^3 \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*Log[c*(d + e*x^(2/3))^n]),x]

[Out]

(-2*b*d^4*n*x^(1/3))/(3*e^4) + (2*b*d^3*n*x)/(9*e^3) - (2*b*d^2*n*x^(5/3))/(15*e^2) + (2*b*d*n*x^(7/3))/(21*e)
- (2*b*n*x^3)/27 + (2*b*d^(9/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/(3*e^(9/2)) + (x^3*(a + b*Log[c*(d + e*x
^(2/3))^n]))/3

Rule 2455

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[((f*x)^(m
+ 1)*(a + b*Log[c*(d + e*x^n)^p]))/(f*(m + 1)), x] - Dist[(b*e*n*p)/(f*(m + 1)), Int[(x^(n - 1)*(f*x)^(m + 1))
/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]

Rule 341

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Dist[k, Subst[Int[x^(k*(
m + 1) - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, m, p}, x] && FractionQ[n]

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \, dx &=\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac{1}{9} (2 b e n) \int \frac{x^{8/3}}{d+e x^{2/3}} \, dx\\ &=\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac{1}{3} (2 b e n) \operatorname{Subst}\left (\int \frac{x^{10}}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac{1}{3} (2 b e n) \operatorname{Subst}\left (\int \left (\frac{d^4}{e^5}-\frac{d^3 x^2}{e^4}+\frac{d^2 x^4}{e^3}-\frac{d x^6}{e^2}+\frac{x^8}{e}-\frac{d^5}{e^5 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{2 b d^4 n \sqrt [3]{x}}{3 e^4}+\frac{2 b d^3 n x}{9 e^3}-\frac{2 b d^2 n x^{5/3}}{15 e^2}+\frac{2 b d n x^{7/3}}{21 e}-\frac{2}{27} b n x^3+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac{\left (2 b d^5 n\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{3 e^4}\\ &=-\frac{2 b d^4 n \sqrt [3]{x}}{3 e^4}+\frac{2 b d^3 n x}{9 e^3}-\frac{2 b d^2 n x^{5/3}}{15 e^2}+\frac{2 b d n x^{7/3}}{21 e}-\frac{2}{27} b n x^3+\frac{2 b d^{9/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{3 e^{9/2}}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0976569, size = 135, normalized size = 1.04 \[ \frac{a x^3}{3}+\frac{1}{3} b x^3 \log \left (c \left (d+e x^{2/3}\right )^n\right )-\frac{2 b d^2 n x^{5/3}}{15 e^2}-\frac{2 b d^4 n \sqrt [3]{x}}{3 e^4}+\frac{2 b d^3 n x}{9 e^3}+\frac{2 b d^{9/2} n \tan ^{-1}\left (\frac{\sqrt{e} \sqrt [3]{x}}{\sqrt{d}}\right )}{3 e^{9/2}}+\frac{2 b d n x^{7/3}}{21 e}-\frac{2}{27} b n x^3 \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*Log[c*(d + e*x^(2/3))^n]),x]

[Out]

(-2*b*d^4*n*x^(1/3))/(3*e^4) + (2*b*d^3*n*x)/(9*e^3) - (2*b*d^2*n*x^(5/3))/(15*e^2) + (2*b*d*n*x^(7/3))/(21*e)
+ (a*x^3)/3 - (2*b*n*x^3)/27 + (2*b*d^(9/2)*n*ArcTan[(Sqrt[e]*x^(1/3))/Sqrt[d]])/(3*e^(9/2)) + (b*x^3*Log[c*(
d + e*x^(2/3))^n])/3

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Maple [F]  time = 0.331, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b\ln \left ( c \left ( d+e{x}^{{\frac{2}{3}}} \right ) ^{n} \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*ln(c*(d+e*x^(2/3))^n)),x)

[Out]

int(x^2*(a+b*ln(c*(d+e*x^(2/3))^n)),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*(d+e*x^(2/3))^n)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.93991, size = 806, normalized size = 6.2 \begin{align*} \left [\frac{315 \, b e^{4} n x^{3} \log \left (e x^{\frac{2}{3}} + d\right ) + 315 \, b e^{4} x^{3} \log \left (c\right ) - 126 \, b d^{2} e^{2} n x^{\frac{5}{3}} + 315 \, b d^{4} n \sqrt{-\frac{d}{e}} \log \left (\frac{e^{3} x^{2} - 2 \, d e^{2} x \sqrt{-\frac{d}{e}} - d^{3} + 2 \,{\left (e^{3} x \sqrt{-\frac{d}{e}} + d^{2} e\right )} x^{\frac{2}{3}} - 2 \,{\left (d e^{2} x - d^{2} e \sqrt{-\frac{d}{e}}\right )} x^{\frac{1}{3}}}{e^{3} x^{2} + d^{3}}\right ) + 210 \, b d^{3} e n x - 35 \,{\left (2 \, b e^{4} n - 9 \, a e^{4}\right )} x^{3} + 90 \,{\left (b d e^{3} n x^{2} - 7 \, b d^{4} n\right )} x^{\frac{1}{3}}}{945 \, e^{4}}, \frac{315 \, b e^{4} n x^{3} \log \left (e x^{\frac{2}{3}} + d\right ) + 315 \, b e^{4} x^{3} \log \left (c\right ) - 126 \, b d^{2} e^{2} n x^{\frac{5}{3}} + 630 \, b d^{4} n \sqrt{\frac{d}{e}} \arctan \left (\frac{e x^{\frac{1}{3}} \sqrt{\frac{d}{e}}}{d}\right ) + 210 \, b d^{3} e n x - 35 \,{\left (2 \, b e^{4} n - 9 \, a e^{4}\right )} x^{3} + 90 \,{\left (b d e^{3} n x^{2} - 7 \, b d^{4} n\right )} x^{\frac{1}{3}}}{945 \, e^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*(d+e*x^(2/3))^n)),x, algorithm="fricas")

[Out]

[1/945*(315*b*e^4*n*x^3*log(e*x^(2/3) + d) + 315*b*e^4*x^3*log(c) - 126*b*d^2*e^2*n*x^(5/3) + 315*b*d^4*n*sqrt
(-d/e)*log((e^3*x^2 - 2*d*e^2*x*sqrt(-d/e) - d^3 + 2*(e^3*x*sqrt(-d/e) + d^2*e)*x^(2/3) - 2*(d*e^2*x - d^2*e*s
qrt(-d/e))*x^(1/3))/(e^3*x^2 + d^3)) + 210*b*d^3*e*n*x - 35*(2*b*e^4*n - 9*a*e^4)*x^3 + 90*(b*d*e^3*n*x^2 - 7*
b*d^4*n)*x^(1/3))/e^4, 1/945*(315*b*e^4*n*x^3*log(e*x^(2/3) + d) + 315*b*e^4*x^3*log(c) - 126*b*d^2*e^2*n*x^(5
/3) + 630*b*d^4*n*sqrt(d/e)*arctan(e*x^(1/3)*sqrt(d/e)/d) + 210*b*d^3*e*n*x - 35*(2*b*e^4*n - 9*a*e^4)*x^3 + 9
0*(b*d*e^3*n*x^2 - 7*b*d^4*n)*x^(1/3))/e^4]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*ln(c*(d+e*x**(2/3))**n)),x)

[Out]

Timed out

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Giac [A]  time = 1.33877, size = 140, normalized size = 1.08 \begin{align*} \frac{1}{3} \, b x^{3} \log \left (c\right ) + \frac{1}{3} \, a x^{3} + \frac{1}{945} \,{\left (315 \, x^{3} \log \left (x^{\frac{2}{3}} e + d\right ) + 2 \,{\left (315 \, d^{\frac{9}{2}} \arctan \left (\frac{x^{\frac{1}{3}} e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{11}{2}\right )} -{\left (315 \, d^{4} x^{\frac{1}{3}} e^{4} - 105 \, d^{3} x e^{5} + 63 \, d^{2} x^{\frac{5}{3}} e^{6} - 45 \, d x^{\frac{7}{3}} e^{7} + 35 \, x^{3} e^{8}\right )} e^{\left (-9\right )}\right )} e\right )} b n \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*(d+e*x^(2/3))^n)),x, algorithm="giac")

[Out]

1/3*b*x^3*log(c) + 1/3*a*x^3 + 1/945*(315*x^3*log(x^(2/3)*e + d) + 2*(315*d^(9/2)*arctan(x^(1/3)*e^(1/2)/sqrt(
d))*e^(-11/2) - (315*d^4*x^(1/3)*e^4 - 105*d^3*x*e^5 + 63*d^2*x^(5/3)*e^6 - 45*d*x^(7/3)*e^7 + 35*x^3*e^8)*e^(
-9))*e)*b*n